Derivation of rate equations

Derivation of rate equations is an integral part of the effective usage of kinetics as a tool. Novel mechanisms must be described by new equations, and famihar ones often need to be modified to account for minor deviations from the expected pattern. The mathematical manipulations involved in deriving initial velocity or isotope exchange-rate laws are in general quite straightforward, but can be tedious. It is the purpose of this entry, therefore, to present the currently available methods with emphasis on the more convenient ones. 

This use of the determinant method for complex enzyme mechanisms is time-consuming because of the stepwise expansion and the large number of positive and negative terms that must be canceled. It is quite useful, however, in computer-assisted derivation of rate equations. ... 

The Method of Volkenstein and Goldstein. Volkenstein and Goldstein have applied the theory of graphs to the derivation of rate equations. Their approach has three main features the use of an auxiliary node, the compression of a path into a point, and the addition of parallel branches. These can be best explained by an example (Scheme 3). 

The systematic method is equally convenient for the derivation of rate equations for simple mechanisms. Scheme 1, for example, can be redrawn as an enclosed figure after deleting the pathways between unlabeled enzyme forms. 

For catalytic cycles with more than three or four members, the long-hand derivation of rate equations gets out of hand. However, a general formula comparable to that given in Section 6.3 for noncatalytic simple pathways was established as early as 1931 by Christiansen [36-38]. ... 

In copolymerization, several different combinations of initiation and termination mechanisms are possible, giving rise to a variety of different polymerization rate equations. Only two cases will be singled out here free-radical copolymerization with termination by coupling, and ionic polymerization with termination by chain transfer to a deactivating agent or impurity. For other combinations, the derivation of rate equations follows along the same lines. 

This is the central equation to start the derivation of rate equations, and we will see below how this equation can be used for various quantities X. 

Ishikawa, H., Maeda, T., Hikita, H and Miyatake, K. (1988) The computerised derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption. Biochem. J. 251, 175-181. 

It is clear that the material given in this chapter is quite classical and has been known in the literature since the 1930s and 1940s in the field of surface chemistry and catalysis. In fact this is the extent of knowledge used to date in the derivation of rate equations for gas solid catalytic reactions. To be more specific most of the studies on the development of gas-solid catalytic reactions do not even use the information and knowledge related to the rates of chemisorption (activated or non-activated) and desorption. Even the most detailed kinetic studies, usually rely on the assumption of equilibrium adsorption-desorption and use one of the well known equilibrium isotherms (usually the Langmuir isotherm) in order to relate the surface concentration to the concentration of the gas just above the surface of the catalyst. 

The derivation of rate equations for simultaneous hydrogenation of TMP-aldol and formaldehyde was based on a plausible surface reaction mechanism. According to the mechanism, the aldol, formaldehyde and hydrogen undergo competitive adsorption on the nickel-alumina surface. Adsorbed hydrogen atoms add to the carbonyl bonds of the aldol and formaldehyde. These irreversible reaction steps are presumed to be rate-determining, whereas the product desorption is regarded as rapid. Consequently, the reaction mechanism is written as follows ... 

The same strategy was applied in the derivation of rate equations for w-step nucleation according to a power law (cf. Eq. (21)) [133, 134], the combination of nucleation laws with anisotropic growth regimes [153], as well as truncated nucleation due to time-dependent concentration gradients of monomers [136]. MC simulations verified that the Avrami theorem is valid for instantaneous [184], progressive [185], and n-step nucleation according to a power law [184-187]. 

Dimensions are the currency of science. Therefore, it is of utmost importance, especially in enzyme kinetics, to take care of dimensions and to perform the dimensional analysis of kinetic equations and expressions in order to avoid the algebraic mistakes and to check the derivation of rate equations. 

The derivation of rate equations for simple monosubstrate reactions was described in Chapter 3. For bisubstrate reactions, the derivation is usually much more complex, and requires the application of a special mathematical apparatus and special mathematical procedures. 

The interaction factors a, p, and y represent, respectively, the effect of A on the binding of B, the effect of I on the binding of B, and the effect of I on the binding of A. The factor d represents the effect of I on the catalytic activity of the EABI complex. The derivation of rate equation for reaction (6.14) would be extremely laborious from the steady-state assumptions. Therefore, the general velocity equation is derived from Ae rapid equilibrium assumptions ... 

In this section, we shall reviewthe rate equations forthe majortypes of trisubstrate mechanisms, written in the absence of products (Cleland, 1963 Plowman, 1972 Fromm, 1975,1979). All trisubstrate mechanisms in the rapid equilibrium category are relatively rare and the steady-state mechanisms are more common. However, the derivation of rate equations for rapid equilibrium mechanisms, in the absence of products, is less demanding, as it requires only the rapid equilibrium assumptions and, therefore, the resulting rate equations are relatively simple. 

If the rates of proton-transfer steps are high, in specific cases, the protons can be introduced into the rate equations for bisubstrate and even trisubstrate reactions in a straightforward manner, without a need for a complete derivation of rate equations (Schulz, 1994). If the protons are treated as dead-end inhibitors of enzymatic reactions, the concentration terms for protons can be introduced directly into the rate equations via the enzyme distribution equations, alleviating considerably the derivation procedure. In order to illustrate this type of analysis, consider the usual Ordered Bi Bi mechanism (Section 9.2). 

The derivation of rate equations for isotope exchange away from equilibrium may be understood in terms of Eq. (16.7) for the Ordered Bi Bi mechanism... 

Let us start the derivation of rate equations for kinetic isotope effects with analysis of a simple monosubstrate enzyme reaction ... 

Equations (17.67)-(17.69) are based on the assumption that only the rate constants of the chemical step are subject to a kinetic solvent isotope effect. This assumption simplifies the derivation of rate equations, but one must always keep in mind that, though the substrate binding and product release are usually solvent isotope-insensitive, the entire mechanism may contain more than a single isotope sensitive step. 

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